8b(3b+4)=11

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Solution for 8b(3b+4)=11 equation: 


Simplifying
8b(3b + 4) = 11

Reorder the terms:
8b(4 + 3b) = 11
(4 * 8b + 3b * 8b) = 11
(32b + 24b2) = 11

Solving
32b + 24b2 = 11

Solving for variable 'b'.

Reorder the terms:
-11 + 32b + 24b2 = 11 + -11

Combine like terms: 11 + -11 = 0
-11 + 32b + 24b2 = 0

Begin completing the square.  Divide all terms by
24 the coefficient of the squared term: 

Divide each side by '24'.
-0.4583333333 + 1.333333333b + b2 = 0

Move the constant term to the right:

Add '0.4583333333' to each side of the equation.
-0.4583333333 + 1.333333333b + 0.4583333333 + b2 = 0 + 0.4583333333

Reorder the terms:
-0.4583333333 + 0.4583333333 + 1.333333333b + b2 = 0 + 0.4583333333

Combine like terms: -0.4583333333 + 0.4583333333 = 0.0000000000
0.0000000000 + 1.333333333b + b2 = 0 + 0.4583333333
1.333333333b + b2 = 0 + 0.4583333333

Combine like terms: 0 + 0.4583333333 = 0.4583333333
1.333333333b + b2 = 0.4583333333

The b term is 1.333333333b.  Take half its coefficient (0.6666666665).
Square it (0.4444444442) and add it to both sides.

Add '0.4444444442' to each side of the equation.
1.333333333b + 0.4444444442 + b2 = 0.4583333333 + 0.4444444442

Reorder the terms:
0.4444444442 + 1.333333333b + b2 = 0.4583333333 + 0.4444444442

Combine like terms: 0.4583333333 + 0.4444444442 = 0.9027777775
0.4444444442 + 1.333333333b + b2 = 0.9027777775

Factor a perfect square on the left side:
(b + 0.6666666665)(b + 0.6666666665) = 0.9027777775

Calculate the square root of the right side: 0.950146187

Break this problem into two subproblems by setting 
(b + 0.6666666665) equal to 0.950146187 and -0.950146187.


Subproblem 1
b + 0.6666666665 = 0.950146187

Simplifying
b + 0.6666666665 = 0.950146187

Reorder the terms:
0.6666666665 + b = 0.950146187

Solving
0.6666666665 + b = 0.950146187

Solving for variable 'b'.

Move all terms containing b to the left, all other terms to the right.

Add '-0.6666666665' to each side of the equation.
0.6666666665 + -0.6666666665 + b = 0.950146187 + -0.6666666665

Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000
0.0000000000 + b = 0.950146187 + -0.6666666665
b = 0.950146187 + -0.6666666665

Combine like terms: 0.950146187 + -0.6666666665 = 0.2834795205
b = 0.2834795205

Simplifying
b = 0.2834795205


Subproblem 2
b + 0.6666666665 = -0.950146187

Simplifying
b + 0.6666666665 = -0.950146187

Reorder the terms:
0.6666666665 + b = -0.950146187

Solving
0.6666666665 + b = -0.950146187

Solving for variable 'b'.

Move all terms containing b to the left, all other terms to the right.

Add '-0.6666666665' to each side of the equation.
0.6666666665 + -0.6666666665 + b = -0.950146187 + -0.6666666665

Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000
0.0000000000 + b = -0.950146187 + -0.6666666665
b = -0.950146187 + -0.6666666665

Combine like terms: -0.950146187 + -0.6666666665 = -1.6168128535
b = -1.6168128535

Simplifying
b = -1.6168128535


Solution
The solution to the problem is based on the solutions
from the subproblems.
b = {0.2834795205, -1.6168128535}

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